Feasibility of Ultra–Wideband Channels at Millimeter Wavelengths Faded by Rain in GeoSurf Satellite Constellations

1. Satellite Constellations with Zenith Propagation Paths at Any Site

Among satellite constellations, the GeoSurf constellations are a good choice for future worldwide internet because they share most of the advantages of current GEO (Geostationary), MEO (Medium Earth Orbit), and LEO (Low Earth Orbit) satellite constellations without suffering most of their drawbacks, because they emulate the geostationary orbit with zenith paths in sites located at any latitude [1].

In [2] we have compared the tropospheric attenuation of GeoSurf zenith paths with the paths of GEO, MEO and LEO satellites. In [3] we have assessed how the annual average probability distribution $P\left(A\right)$of exceeding a given rain attenuation $A$ (dB) depends on the carrier frequency $f_c$(GHz) in the GeoSurf paths at sites in different climatic regions. All these studies refer to narrow–band channels, i.e. channels with negligible linear (amplitude and phase) distortions.

In [4] we have estimated the slowly time–varying transfer–function and linear distortions that are likely found in ultra–wideband radio links in GeoSurf constellations working at millimeter wavelengths. As a practical example, the bandwidth considered was 10 GHz wide, centered at 80 GHz (W–Band), because we think it might be used in future worldwide internet radio links using spread spectrum modulation and Code Division Multiple Access (CDMA) [5–10] with BPSK and QPSK modulation, once high–frequency large wideband technology – now developed at lower frequencies [11] – will also be available at W–Band. CDMA can provide a large processing gain and can be designed without considering issues related to frequency and access coordination which, together with the advantages of the GeoSurf constellations, can be an effective choice to provide high bit rates to users.

The literature on what today is defined “wideband” communication does not refer to rain attenuation, or to the ultra–wideband radio links here studied, but only to radio links in clear–sky (mainly multipath), both for terrestrial and satellite systems [5, 12–16], therefore here we further discuss these topics, after [4] and [17].

Following our previous study [4], the purpose of this paper is to assess – with simulations using on–site rain–rate time series and the Synthetic Storm Technique (SST) [18] – the interference due to amplitude and phase distortions produced by rain attenuation in ultra–wideband communication systems designed for using double sideband–suppressed carrier (DSB–SC) modulation in both quadrature channels (QPSK). We assume that the total signal–to–noise ratio (SNR) of the Gaussian channel is greater than the minimum required to guarantee a bit error probability smaller than the maximum value tolerated by users. Under this hypothesis, we estimate how the in–band attenuation and phase delay affect the baseband digital signal. We evaluate the sampler output of a direct channel (e.g., the cosine channel, inter–symbol interference, ISI) and the quadrature interference (QI) coming from the orthogonal channel (the sine channel, referred to also as quadrature channel).

For illustrating the characteristics of the interference, we report the results concerning radio–links simulated at Spino d’Adda (Italy), Madrid (Spain) and Tampa (Florida), sites in different climatic regions (Table 1). We have considered these specific sites because rain‒rate time series $R\left(t\right)$ (mm/h) – averaged in 1‒min intervals – have been continuously recorded on‒site for several years, a sufficiently long period to provide reliable experimental results with SST.

The rain fade in GeoSurf radio links is independent of the particular GeoSurf design (e.g., altitude and number of satellites), because all paths to/from a satellite of the constellation are always locally vertical (zenith). To determine the slowly time–varying passband and baseband equivalent transfer–functions distorted by rain we must know the rain–attenuation $A\left(t\right)$ (dB) and the phase–delay $\mathsf{\Phi }\left(\mathrm{t}\right)$(degrees) time series. Both are calculated with the SST as shown in [4].

After this introduction, Section 2 shows examples of the results obtained with the SST; Section 3 shows how to calculate passband and baseband transfer–functions; Section 4 reports histograms of interference; Section 5 models probability distributions of interference; Section 6 calculates the theoretical channel capacity loss; Section 7 reports a general conclusion and indicates future work.

2. Rain Attenuation and Phase Delay Due to Rain in Zenith Propagation Paths

To calculate the passband (radio–frequency) and baseband equivalent transfer–functions of direct and orthogonal–channels in DSB–SC modulation, defined and discussed in [4] and based on classical linear modulation theory (e.g. [19, 20]), we must know the time series $A\left(t\right)$ (dB) and t $\mathsf{\Phi }\left(\mathrm{t}\right)$(degrees). Both are calculated with the SST [18], as shown in [4].

Figure 1 shows the average annual probability distribution $P\left(R\right)$ of exceeding $R$ (mm/h, averaged in 1 minute) measured at Spino d’Adda, Madrid and Tampa. The different climatic rain conditions at these sites are clearly evident.

Figure 1. Annual probability distribution (%) $P\left(\mathrm{R}\right)$ of exceeding the value indicated in abscissa, at Spino d’Adda, Madrid and Tampa.

Figure 1. Annual probability distribution (%) $P\left(\mathrm{R}\right)$ of exceeding the value indicated in abscissa, at Spino d’Adda, Madrid and Tampa.

Figure 2 (a) shows the annual probability distribution $P\left(A\right)$ of exceeding $A$(dB) calculated with Equation (1), and Figure 2(b) shows the corresponding annual probability distribution $P\left(\mathsf{\Phi }\right)$ of exceeding $\mathsf{\Phi }$ (degrees) calculated with Equation (2).

Figure 2. (a) Annual probability distribution (%) $P\left(\mathrm{A}\right)$ of exceeding the value indicated in abscissa, at Spino d’Adda, Madrid and Tampa; (b) Annual probability distribution (%) $P\left(\mathsf{\Phi }\right)$ of exceeding the value indicated in abscissa, at Spino d’Adda, Madrid and Tampa, at 80 GHz, circular polarization.

Figure 2. (a) Annual probability distribution (%) $P\left(\mathrm{A}\right)$ of exceeding the value indicated in abscissa, at Spino d’Adda, Madrid and Tampa; (b) Annual probability distribution (%) $P\left(\mathsf{\Phi }\right)$ of exceeding the value indicated in abscissa, at Spino d’Adda, Madrid and Tampa, at 80 GHz, circular polarization.

Now, from $A\left(t\right)$ and $\mathsf{\Phi }\left(t\right)$, the transfer function can be calculated at any time $t$, as we will recall in Section 3. Notice that, since $R\left(t\right)$ is averaged in 1 min, $A\left(t\right)$ and $\mathsf{\Phi }\left(t\right)$ are also “averaged” in 1 min, so that the time–varying transfer function changes every minute, therefore very slowly compared to any digital signal so that in every minute the Gaussian channel is affected by the same in–band rain attenuation and phase delay.

Moreover, we assume that at any time $t$, the channel SNR is larger than the minimum required by the bit error probability constraint, therefore the rain fade to be compensated with power control, or with other techniques, is that measured at the highest frequency of the upper side–band, i.e. at 85 GHz in our exercise. Because our purpose is to study transfer–functions, we assume the channel is still working as required; in other words, outages are not considered as they are not directly related to transfer–functions.

An example illustrates how the SST transforms $R\left(t\right)$into $A\left(t\right)$ and $T\left(t\right)$. Figure 3 shows $R\left(t\right)$recorded in Tampa and Figures 4, 5 show the corresponding $A\left(t\right)$ and $T\left(t\right)$ at 80 GHz and the relative values at the extremes of the 10 GHz bandwidth centered at 80 GHz. Notice that since $A\left(t\right)$ and $T\left(t\right)$ increase with frequency non–linearly in decibels and in picoseconds, the relative differences are not odd functions.

Figure 3. Rain–rate time series $R\left(\mathrm{t}\right)$ recorded at Tampa on 27 September 1997. According to the rain–gauge collecting rainfall, the rain event started at 7:34:27 AM, local time. Samples are averaged in 1 minute.

Figure 3. Rain–rate time series $R\left(\mathrm{t}\right)$ recorded at Tampa on 27 September 1997. According to the rain–gauge collecting rainfall, the rain event started at 7:34:27 AM, local time. Samples are averaged in 1 minute.

Figure 4. SST simulated event at Tampa, 27 September 1997, started at 7:34:27 AM local time. Upper panel: Rain attenuation $A\left(t\right)$ (dB) at 80 GHz (circular polarization). Lower panel: Relative attenuation at the extremes of a 10–GHz bandwidth. Sampling time is 1 minute.

Figure 4. SST simulated event at Tampa, 27 September 1997, started at 7:34:27 AM local time. Upper panel: Rain attenuation $A\left(t\right)$ (dB) at 80 GHz (circular polarization). Lower panel: Relative attenuation at the extremes of a 10–GHz bandwidth. Sampling time is 1 minute.

Figure 5. SST simulated event at Tampa, 27 September 1997, started at 7:34:27 AM, local time. Upper panel: Phase delay $T\left(t\right)$ (ps) at 80 GHz (circular polarization). Lower panel: Relative phase delay at the extremes of a 10–GHz bandwidth. Sampling time is 1 minute.

Figure 5. SST simulated event at Tampa, 27 September 1997, started at 7:34:27 AM, local time. Upper panel: Phase delay $T\left(t\right)$ (ps) at 80 GHz (circular polarization). Lower panel: Relative phase delay at the extremes of a 10–GHz bandwidth. Sampling time is 1 minute.

3. Ultra–Wideband Channels Distorted by Rain

In this section we recall the theoretical flow-chart of the digital receivers. Figures 6 and 7 show the baseband receiver in ideal and in rain conditions. $S\left(f\right)$ is the two–sided spectrum of the Nyquist pulse, $\sqrt{S\left(f\right)}$ is the matched filter, $n\left(t\right)$ is the receiver total additive Gaussian white noise.

Figure 6. Flow-chart of the baseband receiver in ideal conditions. $S\left(f\right)$ is the two–sided spectrum of the Nyquist pulse, $\sqrt{S\left(f\right)}$ is its matched filter, $n\left(t\right)$ is the receiver total additive Gaussian white noise.

Figure 6. Flow-chart of the baseband receiver in ideal conditions. $S\left(f\right)$ is the two–sided spectrum of the Nyquist pulse, $\sqrt{S\left(f\right)}$ is its matched filter, $n\left(t\right)$ is the receiver total additive Gaussian white noise.

Figure 7. Flow-chart of the quadrature baseband receiver in rain attenuation. $S\left(f\right)$ is the two–sided spectrum of the Nyquist reference pulse assumed to be positive, $\sqrt{S\left(f\right)}$ is the matched filter and $n\left(t\right)$ is the receiver total additive Gaussian white noise for each channel, of equal power.

Figure 7. Flow-chart of the quadrature baseband receiver in rain attenuation. $S\left(f\right)$ is the two–sided spectrum of the Nyquist reference pulse assumed to be positive, $\sqrt{S\left(f\right)}$ is the matched filter and $n\left(t\right)$ is the receiver total additive Gaussian white noise for each channel, of equal power.

In the ideal case (Figure 6) the amplitude at the sampling time is normalized to unity. In other words, a hard decision gives $\mp 1$ by sampling the pulse peak at the sampling time $t=0$.

In the rain fade case (Figure 7), we suppose that, at any time $t$, the received signal is amplified by multiplying it by ${10}^{\frac{A\left(f_c\right)}{20}}$ to get rid of the variable rain attenuation at the carrier frequency $f_c=80$ GHz (this is not related to the outage condition discussed above, which must consider the rain attenuation at 85 GHz). After this multiplication, only the in–band signal variations about the carrier frequency are left, which is the effect we want to study. At the output of the sampler, before the hard decision, we find the algebraic sum $s_{sd}+s_{sq}$ of direct and orthogonal–channel samples at the reference time $t=0$, corresponding to the peak value of the direct (cosine) and orthogonal–channel (sine) pulses present simultaneously in the two orthogonal channels. In general, however, both values are affected by self–channel (ISI) interference due to the tails of the previous and following pulses and to the interference coming from the orthogonal–channel (QI).

In the following, we consider the relative phase delay (degrees) at radiofrequency $\psi$:

$$\phi \left(\psi \right)=\mathsf{\Phi }\left(\psi \right)-\mathsf{\Phi }(\psi =f_c)$$

(1)

and the relative time delay:

$$\tau \left(f\right)=T\left(\psi \right)-T(\psi =f_c)$$

(2)

Defined the normalized magnitude $\left|H\left(\psi \right)\right|$ of the passband transfer–function $H\left(\psi \right)$, at time $t,$ as:

$$\left|H\left(\psi \right)\right|={10}^{\frac{A\left(f_c\right)-A\left(\psi \right)}{20}}$$

(3)

the real and imaginary parts of $H\left(\psi \right)$are given by

$$H_r\left(\psi \right)=\left|H\left(\psi \right)\right|\mathrm{c}\mathrm{o}\mathrm{s}\left(\phi \right(\psi \left)\right)$$

(4)

$$H_i\left(\psi \right)=\left|H\left(\psi \right)\right|\mathrm{s}\mathrm{i}\mathrm{n}\left(\phi \right(\psi \left)\right)$$

From these equations, as shown in [4], we can calculate the baseband equivalent transfer–functions at frequency $f$, whose averages for Spino d’Adda, Madrid and Tampa are reported in [4].

In the next Section we report the statistics of interference due to the direct channel (ISI) and to the quadrature channel (QI). These results are fundamental for assessing the feasibility of the ultra–wideband radio link.

4. Experimental Interference

Numerical results have been obtained by means of time–domain simulations of the system described in Figure 9 in MATLAB/Simulink^®. We have simulated the transmission of a sequence of ${10}^4$ symbols drawn from a quadrature phase–shift keying (QPSK) modulation long sequences of independent bits at the bit rate $\rho$ (bits per second, bps) according to the value of the roll–off factor $r$. Since the baseband width is fixed to $B=5$ GHz, the bit rate is a function of $r$, given by:

$$\rho =\frac{2B}{1+r}$$

(5)

We have simulated bit rates corresponding to $r=0.25$, i.e. $\rho =8$ Gbps; $r=0.50$, $\rho =6.67$ Gbps; $r=1$, $\rho =5$ Gbps. A numerical approximation of the continuous–time squared–root Nyquist filter $\sqrt{S\left(f\right)}$ has been implemented by considering a windowing of 16 symbol intervals and oversampling factor of 4 compared to the symbol–rate.

Because we refer to normalized values, the algebraic sum of total interference $s_{sd}+s_{sq}$ gives the factor by which the SNR $\mathsf{\Gamma }$ obtainable in the ideal channel (Figure 8) must be multiplied to obtain the SNR in presence of ISI and QI. Therefore, defined the channel factor $\mu$:

$$\mu =s_{sd}+s_{sq}$$

(6)

the SNR obtainable with interference is given by:

$$\gamma ={\mu }^2\mathsf{\Gamma }$$

(7)

$$\gamma \left(\mathrm{d}\mathrm{B}\right)=10\times {\mathrm{l}\mathrm{o}\mathrm{g}}_{10}{\mu }^2+10\times {\mathrm{l}\mathrm{o}\mathrm{g}}_{10}\mathsf{\Gamma }$$

Figure 8. Relative frequency histogram of the channel factor $\mu$ (dB) with ISI (upper panel) and with ISI and QI (lower panel.). Color key: blue=Spino d’Adda; green=Madrid; red=Tampa. Curve key: continuous $9\le A_{80\mathrm{G}\mathrm{H}\mathrm{z}}\le 11$ dB; dashed $18\le A_{80\mathrm{G}\mathrm{H}\mathrm{z}}\le 22$ dB; “+” $27\le A_{80\mathrm{G}\mathrm{H}\mathrm{z}}\le 33$ dB. Roll–off factor $r=0.25$.

Figure 8. Relative frequency histogram of the channel factor $\mu$ (dB) with ISI (upper panel) and with ISI and QI (lower panel.). Color key: blue=Spino d’Adda; green=Madrid; red=Tampa. Curve key: continuous $9\le A_{80\mathrm{G}\mathrm{H}\mathrm{z}}\le 11$ dB; dashed $18\le A_{80\mathrm{G}\mathrm{H}\mathrm{z}}\le 22$ dB; “+” $27\le A_{80\mathrm{G}\mathrm{H}\mathrm{z}}\le 33$ dB. Roll–off factor $r=0.25$.

In applying these equations, we assume that the channel Gaussian noise power is constant, although it should slightly increase with rain attenuation.

Our main results are the histograms (relative frequency) of the sampler output amplitude (dB) first only with ISI and then with ISI and QI, i.e. with the total interference. To show the dependence on the rain attenuation calculated at center frequency (i.e., 80 GHz), we have considered the cases in which rain attenuation (and corresponding time delay) was in the following ranges: $9\le A_{80\mathrm{G}\mathrm{H}\mathrm{z}}\le 11$ dB (referred to, for short, as “10” dB) $18\le A_{80\mathrm{G}\mathrm{H}\mathrm{z}}\le 22$ dB (“20” dB) and $27\le A_{80\mathrm{G}\mathrm{H}\mathrm{z}}\le 33$ dB (“30” dB).

Figure 8 shows the relative frequency histogram of the channel factor $\mu$ (dB) for $r=0.25$, Figure 9 for $r=0.50$ and Figure 10 with $r=1$.

Figure 9. Relative frequency histogram of the channel factor $\mu$ (dB) with ISI (upper panel) and with ISI and QI (lower panel.). Color key: blue=Spino d’Adda; green=Madrid; red=Tampa. Curve key: continuous $9\le A_{80\mathrm{G}\mathrm{H}\mathrm{z}}\le 11$ dB; dashed $18\le A_{80\mathrm{G}\mathrm{H}\mathrm{z}}\le 22$ dB; “+” $27\le A_{80\mathrm{G}\mathrm{H}\mathrm{z}}\le 33$ dB. Roll–off factor $r=0.50$.

Figure 9. Relative frequency histogram of the channel factor $\mu$ (dB) with ISI (upper panel) and with ISI and QI (lower panel.). Color key: blue=Spino d’Adda; green=Madrid; red=Tampa. Curve key: continuous $9\le A_{80\mathrm{G}\mathrm{H}\mathrm{z}}\le 11$ dB; dashed $18\le A_{80\mathrm{G}\mathrm{H}\mathrm{z}}\le 22$ dB; “+” $27\le A_{80\mathrm{G}\mathrm{H}\mathrm{z}}\le 33$ dB. Roll–off factor $r=0.50$.

From these results we can draw the following general conclusion:

(1)

The three sites, although in different climatic zones, are practically indistinguishable in all cases.

(2)

All histograms show even symmetry, therefore indicating that for about 50% of the time considered the channel factor can be either $\mathit{\mu }<0$ (dB), therefore $\mathit{\gamma }<\mathsf{\Gamma }$, or $\mathit{\mu }>0$ (dB), therefore $\mathit{\gamma }<\mathsf{\Gamma }$.

(3)

Histograms with only ISI and with ISI+QI are distinguishable.

(4)

With ISI only, the lowest range of attenuation (10 dB) shows more marked peaks at $\mathit{\mu }=0$ dB and $\mathit{\mu }\approx \pm 0.3$ dB. These peaks are largely smoothed when QI is also added.

(5)

The roll–off factor $\mathit{r}$ plays a small role, because the peaks change a little their values, especially for $\mathit{r}=1$.

Because of these general findings, in Figure 11 we have drawn the average relative frequency histograms by distinguishing only the three attenuation ranges. In other words, this figure should provide a “global” frequency distribution of the channel factor in a 10 GHz bandwidth centered at 80 GHz. In the next section we will propose a theoretical approach which can justify these findings.

Figure 11. Average relative frequency histogram of the channel factor $\mu$ (dB) in presence of ISI and QI. Curve key: black $9\le A_{80\mathrm{G}\mathrm{H}\mathrm{z}}\le 11$; blue $18\le A_{80\mathrm{G}\mathrm{H}\mathrm{z}}\le 22$; red $27\le A_{80\mathrm{G}\mathrm{H}\mathrm{z}}\le 33$. Curves are averaged over sites and roll–off factors.

Figure 11. Average relative frequency histogram of the channel factor $\mu$ (dB) in presence of ISI and QI. Curve key: black $9\le A_{80\mathrm{G}\mathrm{H}\mathrm{z}}\le 11$; blue $18\le A_{80\mathrm{G}\mathrm{H}\mathrm{z}}\le 22$; red $27\le A_{80\mathrm{G}\mathrm{H}\mathrm{z}}\le 33$. Curves are averaged over sites and roll–off factors.

5. Modelling Interference

In this section we propose a theory which can justify the relative frequency histogram of the channel factor $\mu$ found in Section 5.

The channel factor $\mu$ at the sampling time $t_k$ is given by:

$$\mu (t_k=0)=\sum _{k=-N}^{k=+N}{s}_{sd}\left(t_k\right)+\sum _{k=-N}^{k=+N}{s}_{sq}\left(t_k\right)$$

$$\mu =i_d+i_q$$

(8)

Now, since both $i_d$ and $i_q$ are stochastic variables given by the sum of many independent values, their probability density function – according to the central limit theorem – should approach the Gaussian distribution, therefore the probability density function of $\mu$ should be given by the convolution of two Gaussian models, i.e. another Gaussian probability distribution with mean value given by the sum of the two mean values and variance given by the sum of the two variances, hypotheisis that holds if all variable are of the same orfder of magnitude [24, 25].

Let us do some useful transformations. According to Figures 8–11, the channel factor $\mu$ ranges between $-1$ dB (i.e., ${10}^{-1/20}=0.891$) and $+1$ dB (${10}^{+1/20}=1.122$). Now, by recalling that $\ln (x)\approx (x-1)$ if $x\approx 1$, we can write:

$$\mu \approx \mathrm{l}\mathrm{n}\left(\mu \right)+1$$

(9)

Therefore, in decibels:

$$\mu \left(\mathrm{d}\mathrm{B}\right)=\frac{10}{\ln \left(10\right)}(\mu -1)$$

(10)

Hence, if $\mu$ is Gaussian, then also $\mu \left(\mathrm{d}\mathrm{B}\right)$ is approximately Gaussian. However, Figures 8–10 seem to suggest two different overlapping Gaussian distributions, especially for $r=1$. In other words, $\mu <1$ and $\mu >1$ seem due to two similar sets of data of equal variance but with mean values set approximately at about $\pm 0.3$ dB, therefore the central limit theorem applies to these two separate distributions. The overall relative frequency histograms shown in Figure 11 confirms this general finding, especially at low fade ($~10$ dB). The peak at about $\mathsf{\mu }=0$ dB can be explained by the Dirac pulse produced by the constant $-\frac{10}{\ln \left(10\right)}$of Equation (14). In the experimental histograms it is due to very small spread of the sampler output amplitude due only to the reference pulse, not to ISI or QI.

Now, the question is: how “bad” is the ultra–wideband channel discussed above? In the next section we attempt a quantitative analysis by using Shannon’s channel capacity.

6. Channel Capacity Loss

In this section we discuss a quantitative analysis of the results obtained in the ultra–wideband channels studied in the previous sections by using Shannon’s channel capacity. For this purpose, let us define the capacity factor $\mathit{c}$, independent of roll–factor and logarithm used in its definition, given by:

$$\mathit{c}=\frac{\mathbf{l}\mathbf{n}(1+{\mathit{\mu }}^2\mathsf{\Gamma })}{\mathbf{l}\mathbf{n}(1+\mathsf{\Gamma })}$$

(11)

In Figure 12 we show $\mathit{c}$ versus $\mathsf{\Gamma }$ (dB), calculated at the extreme values of $\mathit{\mu }$, namely approximately $\pm 1$ dB.

Figure 12. Capacity factor versus versus $\mathsf{\Gamma }$ (dB) at the extreme values of $\mu$, namely $\pm 1$ dB. The experimental values of Figures 10–12 are within these bounds.

Figure 12. Capacity factor versus versus $\mathsf{\Gamma }$ (dB) at the extreme values of $\mu$, namely $\pm 1$ dB. The experimental values of Figures 10–12 are within these bounds.

Let us discuss only the worst case, the red curve referring to negative values of $\mathit{\mu }$ (dB). We can notice that, as the signal–to–noise ratio $\mathsf{\Gamma }$ (dB) in ideal conditions (no rain) increases, the capacity factor increases from $\mathit{c}=0.825$ at $\mathsf{\Gamma }=3$ dB to $\mathit{c}=0.950$ at $\mathsf{\Gamma }=20$ dB, therefore the capacity loss ranges from 17.5% to 5%.

In conclusion, we think that the capacity loss in the worst case is very limited and that rain, therefore, does not cause significant linear distortions in ultra–wideband channels at millimeter waves.

7. Conclusions

Following our previous studies concerning GeoSurf satellite constellations, in the present paper we have studied the interference due to amplitude and phase distortions produced by rain in ultra–wideband communication systems designed for using double sideband–suppressed carrier (DSB–SC) modulation.

For illustrating the general characteristics of linear interference due to rain in ultra–wideband channels, we have reported and discussed the results concerning radio–links simulated with SST at Spino d’Adda (Italy), Madrid (Spain) and Tampa (Florida), which are sites located in different climatic regions. Numerical results have been obtained by means of time–domain simulations of the system described in Figure 9.

After reviewing the classical theory of passband (radio–frequency) and baseband equivalent transfer–functions of direct and orthogonal–channels in DSB–SC modulation, we have calculated them when radio–links are faded by rain.

Defined the self–channel (ISI) interference due to the tails of the previous and following pulses and the interference coming from the orthogonal–channel (QI), we have reached the following general conclusions:

(1)

The three sites considered, although in different climatic zones, are practically indistinguishable in all cases.

(2)

The channel factor $\mathit{\mu }$ can be either $\mathit{\mu }<0$ (dB), therefore $\mathit{\gamma }<\mathsf{\Gamma }$, or $\mathit{\mu }>0$ (dB), therefore $\mathit{\gamma }<\mathsf{\Gamma }$ with equal probability.

(3)

Histograms with only ISI or with ISI+QI are diverse.

(4)

With ISI only, the lowest range of attenuation (10 dB) shows more marked peaks in the relative frequency histograms at $\mathit{\mu }=0$ dB and $\mathit{\mu }\approx \pm 0.3$ dB. These peaks are largely smoothed when QI is added.

(5)

The roll–off factor $\mathit{r}$ plays a small role, because the peaks change only for $\mathit{r}=1$.

We have proposed a statistical approach which can justify these findings.

In conclusion, the channel theoretical capacity loss, even in the worst case, is very limited and rain, therefore, does not cause significant linear distortions in ultra–wideband channels at millimeter waves in QPSK schemes. Future work should be done in simulating higher order QAM ultra-wideband channels to assess whether these channels are also little affected by linear distortions due to rain.